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Unformatted text preview: Chemistry 222 Fall 2010 STATISTICS, ANALYSIS OF EXPERIMENTAL ERROR (2) GAUSSIAN (Normal) DISTRIBUTION The variation in random error is NORMALLY distributed! (meaning that replicate measurements exhibit a bellshape, normal or Gaussian distribution). An infinite number of measurements ! 2 2 / 2 ) ( 2 1 ) ( σ μ π σ = x e x y The normal Gauss distribution: The case where the limits of integration are ± ∞; the error (“the random variable”) must lie somewhere within this range; the probability must be unity! x y a b μ 2 2 / 2 ) ( 2 1 σ μ π σ = x e y Probability density “ y” vs. the magnitude of the result “x”; the x = random variable (from  ∞ to + ∞ ). σ , the width (measured at inflection points) of the bell  of values assumed by x about the mean (the population average µ ). x = µ , σ = 1. The Gaussian (or “bell”) curve: • the probability density y is at its maximum, y = 0.3989 at x = μ and at σ = 1 . • the shaded area under the curve, from x = a to x = b, represents the probability that x is in the range: ∆ x = b  a • the units of the probability density function y(x) are those of 1/x • the units of µ and σ are the same as those of x . The random variable x can assume any value from  ∞ to + ∞ • the Gaussian distribution is a continuous statistical distribution! Harris, page 69, Fig. 42 : lifetime of a set of electric light bulbs (4,768): a histogram. The mean gives the center of the distribution, the standard deviation measures the width of the distribution. The scaling (normalizing) factor. 2 2 / 2 ) ( 2 1 ) ( σ μ π σ = x e x y The normalized Gaussian curve is obtained when the variable x is substituted by a variable z : σ μ = x z The normalized Gaussian curve is a Gaussian curve for μ = 0 and σ = 1. The cumulative distribution function (the probabilities that the random variable is in the range from  ∞ to  z , and from 0 to z ) are given in Table 1. Table 1 is a typical example of a statistical table used for data treatment. Table 1. Cumulative distribution function of the normalized Gaussian distribution (symmetry).  z  P( ∞ ,z ) P(0, z )  z  P( ∞ ,z ) P(0, z ) 0 0.5000 0.0000 0.1 0.4602 0.0398 2.1 0.0179 0.4821 0.2 0.4207 0.0793 2.2 0.0139 0.4861 0.3 0.3821 0.1179 2.3 0.0107 0.4893 0.4 0.3446 0.1554 2.4 0.0082 0.4918 0.5 0.3085 0.1915 2.5 0.0062 0.4938 0.6 0.2742 0.2258 2.6 0.0047 0.4953 0.7 0.2420 0.2580 2.7 0.0035 0.4965 0.8 0.2119 0.2881 2.8 0.0026 0.4974 0.9 0.1841 0.3159 3.9 0.0019 0.4981 1.0 0.1587 0.3413 3.0 0.001350 0.498650 1.1 0.1357 0.3643 3.1 0.000968 0.499032 1.2 1....
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This note was uploaded on 12/28/2010 for the course CHEMISTRY 222 taught by Professor Wieckowski during the Spring '10 term at University of Illinois at Urbana–Champaign.
 Spring '10
 WIECKOWSKI
 Chemistry

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